Tabletop with a configuration having safety features and maximum seating capacity for a given size

ABSTRACT

Surfaces of objects and the shapes thereof, which are two-dimensional, closed, curvilinear and three-dimensional, closed, curviplanar in nature, and which have outer boundaries that are defined according to a mathematical expression, such that the shapes have no discontinuities, irregularities, inflection or transition points about their periphery that result in potentially unsafe or dangerous angular, sharp corners or edges in the contours of the shapes, and are otherwise also generally ergonomically beneficial and aesthetically pleasing, are disclosed. The shapes are useful in providing surfaces for objects such as items of furniture, windows, doors, and floor coverings. The shapes are particularly useful for providing surfaces for tabletops that also maximize the number of individuals, who may be seated around the perimeter of a table, for a table of given dimensions, with each individual to be seated thereat being allocated a predetermined amount of linear space around the periphery of the table.

FIELD OF THE INVENTION

[0001] This invention generally relates to surfaces of objects and theshapes thereof, which are two-dimensional, closed, curvilinear andthree-dimensional, closed, curviplanar in nature, and which have outerboundaries that are defined according to a mathematical expression, suchthat the shapes have no discontinuities, irregularities, inflection ortransition points about their periphery that result in potentiallyunsafe or dangerous angular, sharp corners or edges in the contours ofthe shapes, and are otherwise also generally ergonomically beneficialand aesthetically pleasing. More particularly, the invention relates tosuch shapes as used for items of furniture, windows, doors, and floorcoverings. Still more particularly, the invention relates to such shapesthat have closed, curvilinear outer boundaries and that aresubstantially two-dimensional, with no angular corners or sharp edges intheir contour, for use as tabletops, wherein the number of individuals,who may be seated around the perimeter of a table incorporating such atabletop, is maximized for a table of given dimensions, as measured by atotal continuous linear dimension, with each individual to be seatedthereat being allocated a predetermined amount of linear space aroundthe periphery of the table.

BACKGROUND OF THE INVENTION

[0002] Plane surfaces such as windows, doors, carpets, tables, and thesurface of furniture and other items have previously been designed sothat those surfaces have boundaries which are rectangular, square,circular, elliptical, or a combination of these geometric shapes.

[0003] In general, the closed outer boundary of a geometric shape isdescribed by the equation (1): $\begin{matrix}{{\frac{x^{u}}{A^{u}} + \frac{y^{v}}{B^{v}}} = 1} & (1)\end{matrix}$

[0004] where x and y are coordinates of points on the outer boundary ofthe geometric shape, with reference to x and y axes defining an ordinaryCartesian coordinate (x, y) system; and A and B are the coordinates ofthe points at which the curve described by the above equation intersectsthe x and y axes. Specifically, the curve intersects the positive x axisat (A, 0), the negative x axis at (−A, 0), the positive y axis at (0,B), and the negative y axis at (0, −B). The exponents u and v are thedegrees or orders of the closed curve, and may be any rational numbers,not just integers. If u=v=2 and A=B, the curve is a circle. If u=v=2 andA is not equal to B, the curve is an ellipse. If u=v=∞ and A=B, theclosed curve is a square. Finally, if u=v=∞ and A is not equal to B, theclosed curve is a rectangle.

[0005] Although the circle and ellipse have commonly been used in thesurfaces previously described, they lack surface area present withrespect to comparably dimensioned squares and rectangles, respectively.On the other hand, the square and rectangular shapes, although providingmaximum surface area for a given A and B, are not aesthetically pleasingand have the ergonomic disadvantage of sharp corners which may causeinjury or other discomfort to users.

[0006] Thus, there exists a need for surfaces shaped such that moresurface area is available than the traditional circular or ellipticalshapes, while eliminating the unattractiveness of and the hazard ofsharp corners of square or rectangular shapes.

SUMMARY OF THE INVENTION

[0007] The invention comprises surfaces, such as table tops, withboundaries defined by the general analytical curve (1), shown above, butwith the further restriction that 2<u<10 and 2<v<10 (2). The lower limitof 2 defines a circle (when A=B) or an ellipse (when A B). The upperlimit of 10 has been found empirically to be the limit after which theboundary shape approximates a rectangle or square, albeit with slightlyrounded corners.

BRIEF DESCRIPTION OF THE DRAWINGS

[0008]FIG. 1 is a graphical representation of two comparative closedcurves for the case of A=4 and B=3 where u=v=2 for one curve and u=v =3for the other curve.

[0009]FIG. 2 is a graphical representation of two comparative closedcurves for the case of A=4 and B=3 where u=v=2 for one curve and u=v=4for the other curve.

[0010]FIG. 3 is a graphical representation of two comparative closedcurves for the case of A=4 and B=3 where u=v=2 for one curve and u=v=6for the other curve.

[0011]FIG. 4 shows a plan view of a table top of an elliptical shape(u=v=2) where A=46 inches and B=23 inches, and shows outlines of 23 inchwidth spaces, each space representing the width needed for the seatingof one person, arranged around the table top.

[0012]FIG. 5 shows a plan view of a table top of a shape where u=v=4 andwhere A=46 inches and B=23 inches, and shows outlines of 23 inch widthspaces, each space representing the width needed for the seating of oneperson, arranged around the table top.

[0013]FIG. 6 is a three-dimensional perspective view of the table ofFIG. 6.

DESCRIPTION OF PREFERRED EMBODIMENTS

[0014] Equation (1) can be converted into the following equation bymultiplying both sides by the factor A^(u)B^(v):

B ^(v) x ^(u) +A ^(u) y ^(v) =A ^(u) B ^(v)  (3)

[0015] Setting x=0, one obtains:

A^(y)y^(v)=A^(u)B^(v)  (4)

y=B  (5)

[0016] Letting y=0, one further obtains:

B^(v)x^(u)=A^(u)B^(v)  (6)

x=A  (7)

[0017] Differentiating equation (3) with respect to x results in:$\begin{matrix}{{{B^{v}{ux}^{u - 1}} + {A^{u}{vy}^{v - 1}\frac{y}{x}}} = 0} & (8)\end{matrix}$

$\begin{matrix}{\frac{y}{x} = \frac{{- B^{v}}{ux}^{u - 1}}{A^{u}{vy}^{v - 1}}} & (9)\end{matrix}$

[0018] Evaluating $\frac{y}{x}$

[0019] at x=0, one obtains: $\begin{matrix}{\frac{y}{x}\begin{matrix}{\quad {= 0}} \\{x = 0}\end{matrix}} & (10)\end{matrix}$

[0020] Finally, evaluating $\frac{y}{x}$

[0021] represented by the equation (9) at y=0, one obtains:$\begin{matrix}{\frac{y}{x}\begin{matrix}{\quad {= \infty}} \\{y = 0}\end{matrix}} & (11)\end{matrix}$

[0022] It has been empirically determined that u and v should not beallowed to reach the value of 10.

[0023] In general, the surface area available increases as the degree ofthe curve increases, with the most dramatic increase occurring from theelliptical shape to the third degree shape.

[0024] FIGS. 1-3, respectively, graphically show the closed curves foru=v=3, u=v=4, and u=v=6, where A=4 and B=3 for all of FIGS. 1-3. Theclosed curves are drawn with respect to a Cartesian coordinate system,wherein the x axis is a first axis of symmetry, 2, for the curves, andthe y axis is a second axis of symmetry, 4, for the curves. Being theaxes of a Cartesian coordinate system, the x and y axes areperpendicular to each other. In all of FIGS. 1-3, the closed curve foru=v=2 and A=4 and B=3, which is an ellipse, is also drawn with respectto the x and y axes for the purpose of comparison to the curves foru=v=3, u=v=4, and u=v=6.

[0025] The use of these boundaries of degree three or higher up todegree nine can be applied in a myriad of plane surfaces commonly used.Examples of such applications are tables, doors, windows, carpets, anyplane surface of furniture, or indeed any plane surface of any objectimaginable. Furthermore, although the surface enclosed by such a shapedboundary may not, in fact, be planar, but rather curved in threedimensional space (curviplanar surface), the shape of the boundarydescribed herein can still be used with like advantages for increasedsurface area yet pleasing aesthetic appearance and beneficial ergonomiccharacteristics.

[0026] A specific example of a tabletop with a shape defined by theboundary of a closed curve, with A and B being set at actual physicaldimensions, such that A=46 inches and B=23 inches, and u=v=4 is shown inFIG. 5. Standard individual seating spaces of 23-inch width are arrangedabout the table. As can be seen in FIG. 5, a total of ten such seatingspaces can be arranged around the table. In contrast, a table top with aboundary, described by a closed curve with the same values of A and B asFIG. 5, but with u=v=2 (an ellipse), is shown in FIG. 4. FIG. 4 showsthat only eight 23-inch seating spaces can be arranged comfortably aboutthat table. The 14-inch seating spaces remaining at each relativelycurved end of the table do not afford comfortable seating spaces.

[0027]FIG. 6 is a three-dimensional (3-D) perspective view of the tableaccording to FIG. 5, showing the way in which ten persons maycomfortably be seated.

What is claimed is:
 1. A surface of an object, comprising a shape havinga closed, curvilinear outer boundary, the outer boundary of the shapebeing defined by a curve having a mathematical expression I$\begin{matrix}{{\frac{x^{u}}{A^{u}} + \frac{y^{v}}{B^{v}}} = 1} & (I)\end{matrix}$

wherein: x and y are coordinates of points (x, y) on the curve, asmeasured with reference to x and y axes of a standard Cartesian (x, y)coordinate system, from the origin (0, 0); A and B are respectivelycoordinates of positive x and y intercepts of the curve, as measuredwith reference to the x and y axes of the standard Cartesian coordinatesystem, the positive x and y intercepts having respective coordinatesA=(A, 0) and B=(0, B); and u and v are orders of the curve, wherein uand v are each rational numbers in the range of from 2 to 10, andwherein u and v alternatively are the same or are different.
 2. Thesurface of an object according to claim 1, wherein the object isselected from the group consisting of: an item of furniture; a door; awindow; and a floor covering.
 3. The surface according to claim 2,wherein the object is an item of furniture.
 4. The surface according toclaim 3, wherein the item of furniture is a tabletop.
 5. The surfaceaccording to claim 2, wherein the object is a floor covering.
 6. Thesurface according to claim 5, wherein the floor covering is selectedfrom the group consisting of: a mat; a rug; and a carpet.
 7. The surfaceaccording to claim 5, wherein the floor covering is free laying and notwall-to-wall.
 8. The surface according to claim 1, which is a flatplanar surface existing in two dimensions.
 9. The surface according toclaim 1, which is a curvi-planar surface existing in three dimensions.10. The surface according to claim 1, wherein A=4, B=3, and u=v= fromabout 2 to about
 6. 11. The surface according to claim 1, wherein A=2,B=1, and 2<u=v<10.
 12. A surface for a table top according to claim 1,wherein a relative ratio of the dimensions of A:B is 2:1; A has anabsolute dimension of from about 40 to 50 inches; B has an absolutedimension of from about 20 to 25 inches; and u=v=4.